Problem: Determine how many solutions exist for the system of equations. ${6x-y = -3}$ ${12x+3y = -21}$
Convert both equations to slope-intercept form: ${6x-y = -3}$ $6x{-6x} - y = -3{-6x}$ $-y = -3-6x$ $y = 3+6x$ ${y = 6x+3}$ ${12x+3y = -21}$ $12x{-12x} + 3y = -21{-12x}$ $3y = -21-12x$ $y = -7-4x$ ${y = -4x-7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+3}$ ${y = -4x-7}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.